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A De Bruijn–Erdös theorem for chordal graphs
Laurent Beaudou, Adrian Bondy, Xiaomin Chen, Ehsan Chiniforooshan,
Maria Chudnovsky, Vasek Chvâtal, Nicolas Fraiman, Yori Zwols
To cite this version:
Laurent Beaudou, Adrian Bondy, Xiaomin Chen, Ehsan Chiniforooshan, Maria Chudnovsky, et al.. A De Bruijn–Erdös theorem for chordal graphs. The Electronic Journal of Combinatorics, Open Journal Systems, 2015, 22 (1), pp.1.70. �hal-01263335�
arXiv:1201.6376v1 [math.CO] 30 Jan 2012
A De Bruijn–Erd˝
os theorem
for chordal graphs
Laurent Beaudou (Universit´e Blaise Pascal, Clermont-Ferrand)1
Adrian Bondy (Universit´e Paris 6)2
Xiaomin Chen (Shanghai Jianshi LTD)3
Ehsan Chiniforooshan (Google, Waterloo)4
Maria Chudnovsky (Columbia University, New York)5
Vaˇsek Chv´atal (Concordia University, Montreal)6
Nicolas Fraiman (McGill University, Montreal)7
Yori Zwols (Concordia University, Montreal)8
Abstract
A special case of a combinatorial theorem of De Bruijn and Erd˝os asserts that every noncollinear set of n points in the plane determines at least n distinct lines. Chen and Chv´atal suggested a possible generalization of this assertion in metric spaces with appropriately defined lines. We prove this generalization in all metric spaces induced by connected chordal graphs.
1
Introduction
It is well known that
(i) every noncollinear set of n points in the plane determines at least n distinct lines.
As noted by Erd˝os [11], theorem (i) is a corollary of the Sylvester–Gallai the-orem (asserting that, for every noncollinear set S of finitely many points in
1 laurent.beaudou@ens-lyon.org 2 adrian.bondy@sfr.fr 3 gougle@gmail.com 4 chiniforooshan@alumni.uwaterloo.ca 5 mchudnov@columbia.edu
Partially supported by NSF grants DMS-1001091 and IIS-1117631 6
chvatal@cse.concordia.ca
Canada Research Chair in Combinatorial Optimization 7
nfraiman@gmail.com 8
the plane, some line goes through precisely two points of S); it is also a spe-cial case of a combinatorial theorem proved later by De Bruijn and Erd˝os [10].
Theorem (i) involves neither measurement of distances nor measurement of angles: the only notion employed here is incidence of points and lines. Such theorems are a part of ordered geometry [7], which is built around the ternary relation of betweenness: point b is said to lie between points a and c if b is an interior point of the line segment with endpoints a and c. It is customary to write [abc] for the statement that b lies between a and c. In this notation, a line uv is defined — for any two distinct points u and v — as
{u, v} ∪ {p : [puv] ∨ [upv] ∨ [uvp]}. (1)
In terms of the Euclidean metric dist, we have
[abc] ⇔
a, b, c are three distinct points and dist(a, b) + dist(b, c) = dist(a, c). (2)
In an arbitrary metric space, equivalence (2) defines the ternary relation of
metric betweennessintroduced in [12] and further studied in [1, 3, 8]; in turn,
(1) defines the line uv for any two distinct points u and v in the metric space. The resulting family of lines may have strange properties. For instance, a line can be a proper subset of another: in the metric space with points u, v, x, y, z and
dist(u, v) = dist(v, x) = dist(x, y) = dist(y, z) = dist(z, u) = 1,
dist(u, x) = dist(v, y) = dist(x, z) = dist(y, u) = dist(z, v) = 2,
we have
vy = {v, x, y} and xy = {v, x, y, z}.
Chen [4] proved, using a definition of uv different from (1), that the Sylvester– Gallai theorem generalizes in the framework of metric spaces. Chen and Chv´atal [5] suggested that theorem (i), too, might generalize in this frame-work:
(ii) True or false? Every metric space on n points, where n ≥ 2, either has at least n distinct lines or else has a line that consists of all n points.
They proved that
• every metric space on n points either has at least lg n distinct lines or else has a line that consists of all n points
and noted that the lower bound lg n can be improved to lg n + 1
2lg lg n +
1 2lg
π
2 − o(1).
Every connected undirected graph induces a metric space on its vertex set, where dist(u, v) is defined as the smallest number of edges in a path from vertex u to vertex v. Chiniforooshan and Chv´atal [6] proved that
• every metric space induced by a connected graph on n vertices either
has Ω(n2/7
) distinct lines or else has a line that consists of all n vertices; we will prove that the answer to (ii) is ‘true’ for all metric spaces induced by connected chordal graphs.
Theorem 1. Every metric space induced by a connected chordal graph on n
vertices, where n ≥ 2, either has at least n distinct lines or else has a line
that consists of all n vertices.
For graph-theoretic terminology, we refer the reader to Bondy and Murty[2].
2
The proof
Given an undirected graph, let us write [abc] to mean that a, b, c are three distinct vertices such that dist(a, b) + dist(b, c) = dist(a, c); this is equivalent to saying that b is an interior vertex of a shortest path from a to c.
Lemma 1. Let s, x, y be vertices in a finite chordal graph such that [sxy]. If
sx = sy, then x is a cut vertex separating s and y.
Proof. The set of all vertices u such that dist(s, u) = dist(s, x) separates s
and y. Among all its subsets that separate s and y, choose a minimal one and call it C. Since x is an interior vertex of a shortest path from s to y, it belongs to C. To prove that C includes no other vertex, assume, to the contrary, that C includes a vertex u other than x.
Our graph with C removed has distinct connected components S and Y such that s ∈ S and y ∈ Y ; the minimality of C guarantees that each of its vertices
has at least one neighbour in S and at least one neighbour in Y . Since each of u and x has at least one neighbour in S, there is a path from u to x with at least one interior vertex and with all interior vertices in S. Let P be a shortest such path; note that P has no chords except possibly the chord ux. Similarly, there is a path Q from u to x with at least one interior vertex, and with all interior vertices in Y , that has no chords except possibly the chord ux. The union of P and Q is a cycle of length at least four; since this cycle must have a chord, vertices u and x must be adjacent. In turn, the union of
Q and ux is a chordless cycle, and so Q has precisely two edges. This means
that some vertex v in Y is adjacent to both u and x.
Write i = dist(s, x) and j = dist(x, y). Since all vertices t with dist(s, t) < i belong to S and since v has no neighbours in S, we must have dist(s, v) > i; since dist(x, v) = 1, we conclude that dist(s, v) = i + 1 and that v ∈ sx. Since sx = sy, it follows that v ∈ sy. Since dist(v, x) = 1 and dist(x, y) = j, we have dist(v, y) ≤ j + 1. From dist(s, v) = i + 1, dist(s, y) = i + j,
dist(v, y) ≤ j + 1, i ≥ 1, j ≥ 1, and v ∈ sy, we deduce that dist(v, y) = j − 1.
Since dist(u, v) = 1, it follows that dist(u, y) ≤ j; since dist(s, u) = i and
dist(s, y) = i + j, we conclude that dist(u, y) = j and u ∈ sy. Since
dist(s, u) = i, dist(s, x) = i, and dist(u, x) = 1, we have u 6∈ sx. But
then sx 6= sy, a contradiction.
A vertex of a graph is called simplicial if its neighbours are pairwise adjacent.
Lemma 2. Let s, x, y be three distinct vertices in a finite connected chordal
graph. If s is simplicial and sx = sy, then xy consists of all the vertices of
the graph.
Proof. Since sx = sy, we have y ∈ sx, and so [ysx] or [syx] or [sxy]; since s
is simplicial, [ysx] is excluded; switching x and y if necessary, we may assume that [sxy]. Given an arbitrary vertex u, we have to prove that u ∈ xy. Let
P be a shortest path from s to u and let Q be a shortest path from u to y.
Lemma 1 guarantees that x is a cut vertex separating s and y, and so the concatenation of P and Q must pass through x. This means that [sxu] or [uxy] (or both). If [uxy], then u ∈ xy; to complete the proof, we may assume that [sxu], and so u ∈ sx.
Since sx = sy, we have [usy] or [suy] or [syu]; since s is simplicial, [usy] is excluded. If [suy], then [sxu] implies [xuy]; if [syu], then [sxy] implies [xyu]; in either case, u ∈ xy.
Proof of Theorem 1. Consider a connected chordal graph on n vertices where n ≥ 2. By a theorem of Dirac [9], this graph has at least two simplicial vertices; choose one of them and call it s. We may assume that the lines sz with z 6= s are pairwise distinct (else some line consists of all n vertices by Lemma 2). Since the graph is connected and has at least two vertices, s has at least one neighbour; choose one and call it u. If u is the only neighbour of s, then every path from s to another vertex must pass through u, and so su consists of all n vertices. If s has a neighbour v other than u, then line uv is distinct from all of the n − 1 lines sz with z 6= s: since s, u, v are pairwise
adjacent, we have s 6∈ uv.
3
Related theorems
In Theorem 1, ‘connected chordal graph’ can be replaced by ‘connected bi-partite graph’:
• every metric space induced by a connected bipartite graph on n vertices, where n ≥ 2, has a line that consists of all n vertices.
In fact, xy consists of all n vertices whenever x and y are adjacent. To prove this, consider an arbitrary vertex u. Since the graph is bipartite, dist(u, x) and dist(u, y) have distinct parities; since dist(x, y) = 1, they differ by at most one. We conclude that dist(u, x) and dist(u, y) differ by precisely one, and so u ∈ xy.
In Theorem 1, ‘connected chordal graph’ can be also replaced by ‘sufficiently large graph of diameter two’: Chiniforooshan and Chv´atal [6] proved that
• every metric space on n points where each nonzero distance equals 1
or 2 has Ω(n4/3
) distinct lines and this bound is tight.
Acknowledgment
The work whose results are reported here began at a workshop held at Con-cordia University in June 2011. We are grateful to the Canada Research Chairs program for its generous support of this workshop. We also thank Luc Devroye, Fran¸cois Genest, and Mark Goldsmith for their participation in the workshop and for stimulating conversations.
References
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[5] X. Chen and V. Chv´atal, Problems related to a de Bruijn–Erd˝os theorem,
Discrete Applied Mathematics 156(2008), 2101–2108.
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